
where \(\gamma \)  is the photon loss rate and \(J\)  the coupling between
the nearest-neighbor cavities [1]}, [2]}. The
photons thermalize thanks to repeated absorption and emission by the
dye molecules with respective rate coefficients \(B_{12}\)  and
\(B_{21}\) . The ground (excited) molecular state occupation is denoted
by \(M_{1(2)}\)  and obeys \(M_1(\mathbf {x})+M_2(\mathbf {x})=M\) , where \(M\)  is the
number of dye molecules at each lattice site. The emission and
absorption coefficients of the thermalized molecules satisfy the the
Kennard-Stepanov relation [3]}, [4]}, [5]}
\(B_{12} = e^{\beta \Delta } B_{21}\)  where \(\Delta \)  is the detuning
between the cavity frequency and the dye zero-phonon transition
frequency.  The Kennard-Stepanov relation for the absorption and
emission coefficients leads to energy relaxation with dimensionless
strength \(\kappa = B_{12} \bar{M}_1/(2T)\)  (we set the Boltzmann
constant \(k_B=1\) ) [6]}.
